Course: Math 269A, Advanced Numerical Analysis

Text: A. Iserles, A first course in the Numerical Analysis of Differential Equations", Cambridge Texts in Applied Mathematics, Cambridge University Press, 1998

Schedule of Lectures:
 Notations and terminology for ODE's and systems of ODE's; reduction of higher order ODE's to 1st order systems of ODE's; the fundamental existence and uniqueness thm. for ODE's (Lipschitz condition) Introduction of Euler's method, order of Euler's method, one step methods (introduction, definition, consistency, local truncation error) Explicit Runge-Kutta (ERK) methods (introduction of the method in the general case, notations in the general case, derivation of ERK of second order); Runge-Kutta method of fourth order Examples of implicit methods: trapezoidal rule, midpoint rule, the theta method, and the implicit Euler's method; computation of orders for these methods Convergence of one-step methods (the general case; see also convergence for Euler's method, etc) Asymptotic expansions for the global discretization error for one step methods, and applications to error estimate Practical implementation of one step methods Linear Multistep methods: examples, derivation using the Lagrange interpolation polynomial Linear multistep methods: definition and computations of the local truncation error, order of the method, consistency Implicit and explicit linear multistep methods, predictor-corrector methods Examples of consistent multistep methods which diverge Linear difference equations: stability (root) condition, general solution Convergence Thm. for linear multistep methods Order and consistency for linear multistep methods Adaptive methods for one-step and multi-step methods, error control, Milne device, extrapolation Stiff differential equations, stability and intervals (regions) of absolute stability, A-stable methods, BDF methods Numerical methods and stability for systems of ODE's Finite difference methods for linear BVP Functional (fixed point) iteration and Newton's iteration for solving systems of ODE's using an implicit method